Advection-diffusion Partial Differential Equation Research Articles

Structural identifiability analysis of linear reaction–advection–diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction–advection–diffusion PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, consideration of spatial analogues of non-spatial models cannot exacerbate structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

Modeling the interplay of oscillatory synchronization and aggregation via cell–cell adhesion

We present a model of systems of cells with intracellular oscillators (‘clocks’). This is motivated by examples from developmental biology and from the behavior of organisms on the threshold to multicellularity. Cells undergo random motion and adhere to each other. The adhesion strength between neighbors depends on their clock phases in addition to a constant baseline strength. The oscillators are linked via Kuramoto-type local interactions. The model is an advection-diffusion partial differential equation with nonlocal advection terms. We demonstrate that synchronized states correspond to Dirac-delta measure solutions of a weak version of the equation. To analyze the complex interplay of aggregation and synchronization, we then perform a linear stability analysis of the incoherent, spatially uniform state. This lets us classify possibly emerging patterns depending on model parameters. Combining these results with numerical simulations, we determine a range of possible far-from equilibrium patterns when baseline adhesion strength is zero: There is aggregation into separate synchronized clusters with or without global synchrony; global synchronization without aggregation; or unexpectedly a ‘phase wave’ pattern characterized by spatial gradients of clock phases. A 2D Lattice-Gas Cellular Automaton model confirms and illustrates these results.

Macrophages Trajectories Smoothing by Evolving Curves

Abstract When analyzing cell trajectories, we often have to deal with noisy data due to the random motion of the cells and possible imperfections in cell center detection. To smooth these trajectories, we present a mathematical model and numerical method based on evolving open-plane curve approach in the Lagrangian formulation. The model contains two terms: the first is the smoothing term given by the influence of local curvature, while the other attracts the curve to the original trajectory. We use the flowing finite volume method to discretize the advection-diffusion partial differential equation. The PDE includes the asymptotically uniform tangential redistribution of curve grid points. We present results for macrophage trajectory smoothing and define a method to compute the cell velocity for the discrete points on the smoothed curve.

Three-Dimensional Analytical Mathematical Diffusion Model of Air Pollutant in a Mixing Layer with Chemical Reaction and Wet Deposition of Larger Particles of Pollutants Due to a Point Source

A three dimensional analytical mathematical diffusion model of air pollutant in a mixing layer with chemical reaction and wet deposition of larger particles of pollutants due to a point source at is furnished to investigate the primary pollutant concentration in a protected sector and it is above the earth surface layer of the earth’s atmosphere. The pollutants are assumed to be emitted only from a ceaseless source in surface layer. The physical phenomenon of the above problem is reduced to advection diffusion partial differential equation and is solved by using the method of Fourier’s technique. The effect of rate of chemical reaction of pollutants and rate of wet deposition of pollutants on the concentration of dispersed pollutants is investigatedfor both stable and neutral atmospheric situations. Concentration contours are plotted and the results are analysed for air pollutants in stable and neutral atmospheric situations. In the upper part of the mixing layer pollutants concentration is less in stable atmospheric condition for smaller roughness length and moderate geostrophic wind speed.

Simulation of Fixed-Bed Chromatographic Processes Considering the Nonlinear Adsorption Isotherms.

This paper presents the numerical approximation of a nonlinear equilibrium-dispersive (ED) model of multicomponent mixtures for simulating single-column chromatographic processes. Using Danckwerts boundary conditions (DBCs), the ED is studied for both generalized and standard bi-Langmuir adsorption isotherms. Advection-diffusion partial differential equations are used to represent fixed-bed chromatographic processes. As the diffusion term is significantly weaker than the advection term, sophisticated numerical techniques must be applied for solving such model equations. In this study, the model equations are numerically solved by using the Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The technique is designed to handle sudden changes (sharp discontinuities) in solutions and to produce highly accurate results. The method is tested with several case studies considering different parameters, and its results are compared with the high-resolution finite volume scheme. One-, two-, and three-component liquid chromatography elutions on fixed beds are among the case studies being considered. The dynamic model and its accompanying numerical case studies provide the initial step toward continuous monitoring, troubleshooting, and effectively controlling the chromatographic processes.

Atmospheric Dispersion Modeling Using a Stochastic Wind Model

In this work, we propose a stochastic wind field based on the Bayesian dynamic linear model to account for the wind flow field in the transient advection-diffusion partial differential equation (PDE). The resulting dispersion model accounts for the time variation in the wind field and meteorological variables, allowing the simulation of a transient regime. The main advantage of using such a wind field model over a Fourier series to fit wind time series is its potential to make predictions. In addition, a suitable methodology is necessary to solve the resulting dispersion model. In this work, we use a finite element formulation appropriate to solve transient advection-diffusion PDEs. We verify the accuracy of the proposed methodology by reproducing a case study considering a field tracer experiment. The model evaluation against experimental data shows the good performance of the proposed dispersion model.

Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection

Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.

Bayesian inference and wind field statistical modeling applied to multiple source estimation

We present a methodology to identify multiple pollutant sources in the atmosphere that combines a data-driven dispersion model with Bayesian inference and uncertainty quantification. The dispersion model accounts for a realistic wind field based on the output of a multivariate dynamic linear model (DLM), estimated from measured wind components time series. The forward problem solution, described by an adjoint transient advection-diffusion partial differential equation, is then obtained using an appropriately stabilized finite element formulation. The Bayesian inference tool accounts for uncertainty in the concentration data and automatically states the balance between the prior and the likelihood. The source parameters are estimated by a Metropolis in Gibbs Monte Carlo Markov chain (MCMC) algorithm with adaptive steps. The MCMC algorithm is initialized with a maximum a posteriori estimator obtained with particle swarm optimization to accelerate convergence. Finally, the proposed methodology seems to outperform inversion techniques from previous works.

A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations

Within the framework of parameter dependent Partial Differential Equations (PDEs), we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n n -width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n n -width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical Principal Orthogonal Decomposition (POD)-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Solving partial differential equation based on extreme learning machine

In this paper, we present a novel learning method based on extreme learning machine algorithm called ELMNET for solving partial differential equations (PDEs). A loss function that relies on partial differential equation (PDE), initial and boundary condition (I/BC) residual was proposed. The proposed loss function is discretization-free and highly parallelizable. The network parameters are determined by solving a system of linear equations using the ELM algorithm. We demonstrated the performance of ELMNET in solving the advection–diffusion PDE (AD-PDE) as case-studies. The experimental results from the proposed method were compared to the efficient deep neural network and they showed that the ELMNET attains significant improvements in term of both accuracy and training time.

Delay-Adaptive Predictor Feedback Control of Reaction–Advection–Diffusion PDEs With a Delayed Distributed Input

We consider a system of reaction–advection–diffusion partial differential equation (PDE) with a distributed input subject to an unknown and arbitrarily large time delay. Using Lyapunov technique, we derive a delay-adaptive predictor feedback controller to ensure the global stability of the closed-loop system in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> sense. More precisely, we express the input delay as a 1-D transport PDE with a spatial argument leading to the transformation of the time delay into a spatially distributed shift. For the resulting mixed transport and reaction–advection–diffusion PDE system, we employ a PDE backstepping design and certainty equivalence principle to derive the suitable adaptive control law that compensates for the effect of the unknown time delay. Our controller ensures the global stabilization in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> sense. Our result is the first delay-adaptive predictor feedback controller with a PDE plant subject to a delayed distributed input. The feasibility of the proposed approach is illustrated by considering a mobile robot that spread a neutralizer over a polluted surface to achieve efficient decontamination with an unknown actuator delay arising from the noncollocation of the contaminant diffusive process and the moving neutralizer source. Consistent simulation results are presented to prove the effectiveness of the proposed approach.

A heterogeneous non-overlapping domain decomposition explicit finite volume method for a real-time hybrid process-state estimator of 3D unsteady advection-diffusion fields

A heterogeneous, nonoverlapping domain-decomposition, explicit, finite volume method (HT-NODDE-FVM) is developed and applied to a real-time, hybrid, process-state estimator of 3D unsteady, advection-diffusion concentration fields produced by stationary or moving gas sources. The hybrid Luenberger-naive estimator uses a single, pointwise concentration measurement of the field taken onboard a sensing aerial vehicle (SAV) and computes in real-time the concentration field in the entire domain, while guiding the SAV to locations that provide optimal information to the estimator. The computational domain of interest is divided into multiple non-overlapping subdomains using a structured and uniform grid. A Luenberger estimator (or observer) in the form of a 3D advection-diffusion partial differential equation (PDE) is developed for the subdomain where the SAV resides, and a naive estimator of a similar form is developed for the remaining subdomains. The transmission conditions are used on the interfaces between adjacent subdomains for data communication. The hybrid Luenberger-naive estimator is discretized in space by the HT-NODDE-FVM with Total Variation Diminishing (TVD) and the resulting semi-discrete equations are integrated by a fourth order Runge-Kutta scheme. Continuity and flux balance transmission conditions are enforced at the interfaces of adjacent subdomains. The HT-NODDE-FVM hybrid estimator is implemented in parallel using OpenMP. Verification and error analysis of the NODDE-FVM are performed with benchmark tests that include the 3D advection PDE for various initial density distributions and the 3D advection-diffusion PDE for instantaneous gas releases under constant wind speed and eddy diffusivities for a range of Peclet numbers. Additional benchmark tests provide verification and error analysis of the HT-NODDE-FVM hybrid estimator for an instantaneous release by a stationary gas source in a large domain with constant atmospheric properties. The tests examine the impact of grid resolution, sensor model, estimation gain, and numerical data, on the L1, L2, and L∞ norms of the estimation error. Parallelization efficiency analysis of the OpenMP implementation of the hybrid estimator is also presented. The hybrid estimator is applied to the simulation of gaseous plumes from stationary and moving sources in km-scale computational domains under realistic atmospheric conditions and the results show that it achieves real-time estimation of the concentration field.

On UAV source seeking with complex dynamic characteristics and multiple constraints: A cooperative standoff monitoring mode

Aiming at the source-seeking problem with complex dynamic characteristics and multiple constraints in a three-dimensional (3-D) environment, a cooperative standoff monitoring mode that uses unmanned aerial vehicles (UAVs) is proposed in this paper. First, the dynamic pollutant field is modeled by a 3-D advection-diffusion partial differential equation (PDE), and quantifiable indexes are proposed for source seeking around multiple constraints, which consist of UAV kinematics, no-fly zones and cumulative exposure to the high concentration of pollutants. Second, to eliminate the essential gradient information in source seeking of the dependence on known advection and diffusion coefficients in the PDE, UAVs cooperatively estimate the spatial gradient of the pollutant field by the least squares method. Third, a cooperative standoff monitoring mode for source seeking is proposed based on dynamic level set tracking and an interfered fluid dynamical system (IFDS), and receding horizon control (RHC) is introduced to improve the source-seeking effect. Finally, source localization is achieved by the UAVs with the multiple constraints being satisfied, and the effectiveness of the proposed method is validated by the simulation results.

Indirect Optimal Control of Advection-Diffusion Fields through Distributed Robotic Swarms

We consider the problem of optimally guiding a large-scale swarm of underwater vehicles that is tasked with the indirect control of an advection-diffusion environmental field. The microscopic vehicle dynamics are governed by a stochastic differential equation (SDE) with drift. The drift terms model the self-propelled velocity of the vehicle and the velocity field of the currents. In the mean-field setting, the macroscopic vehicle dynamics are governed by a Kolmogorov forward equation in the form of a linear parabolic advection-diffusion partial differential equation (PDE). The environmental field is governed by an advection-diffusion PDE in which the advection term is defined by the fluid velocity field. The vehicles are equipped with on-board actuators that enable the swarm to act as a distributed source in the environmental field, modulated by a scalar control parameter that determines the local source intensity. In this setting, we formulate an optimal control problem to compute the vehicle velocity and actuator intensity fields that drive the environmental field to a desired distribution within a specified amount of time. After proving an existence result for the solution of the optimal control problem, we discretize and solve the problem using the Finite Element Method (FEM). We show through numerical simulations the effectiveness of our control strategy in regulating the environmental field to zero or to a desired distribution in the presence of a double-gyre flow field.

Adaptive Boundary Control for a Certain Class of Reaction–Advection–Diffusion System

Several phenomena in nature are subjected to the interaction of various physical parameters, which, if these latter are well known, allow us to predict the behavior of such phenomena. In most cases, these physical parameters are not exactly known, or even more these are unknown, so identification techniques could be employed in order to estimate their values. Systems for which their inputs and outputs vary both temporally and spatially are the so-called distributed parameter systems (DPSs) modeled through partial differential equations (PDEs). The way in which their parameters evolve with respect to time may not always be known and may also induce undesired behavior of the dynamics of the system. To reverse the above, the well-known adaptive boundary control technique can be used to estimate the unknown parameters assuring a stable behavior of the dynamics of the system. In this work, we focus our attention on the design of an adaptive boundary control for a parabolic type reaction–advection–diffusion PDE under the assumption of unknown parameters for both advection and reaction terms and Robin and Neumann boundary conditions. An identifier PDE system is established and parameter update laws are designed following the certainty equivalence approach with a passive identifier. The performance of the adaptive Neumann stabilizing controller is validated via numerical simulation.

A statically condensed discontinuous Galerkin spectral element method on Gauss-Lobatto nodes for the compressible Navier-Stokes equations

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers.The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advection-diffusion partial differential equations in conservation form with the standard and the split-form variants of the DGSEM. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system.The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summation-by-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

An algorithm for solving the Burgers–Huxley equation using the Elzaki transform

In this research work, we propose an algorithm which involves the coupling of a new integral transform called the Elzaki transform and the well-known homotopy perturbation method on the Burgers–Huxley equation which is a type of nonlinear advection–diffusion partial differential equation. The Burgers–Huxley equation which models reaction mechanisms, diffusion transports and nerve ion propagation which is applicable in traffic flows, acoustics, turbulence theory, hydrodynamics and generally mechanics is the fusion of the well-known Burgers equation and the Huxley equation. We proffer an analytical solution in the form of a Taylor multivariate series of displacement x and time t using the proposed Elzaki homotopy transformation perturbation method (EHTPM) to three cases of the Burgers–Huxley equation. This solution converges rapidly to a closed form which is the same as the exact solutions obtained using the normal analytical methods from the existing literatures. The exact results and that of our proposed EHTPM when compared via tables and 3D plots show an excellent agreement devoid of errors.

Leaderless cooperative control of robotic sensor networks for monitoring dynamic pollutant plumes

In this paper, the problem of cooperative control of robotic sensor networks (RSNs) for monitoring dynamic pollutant plumes in two-dimensional (2D) space is studied. The pollutant plume propagation dynamics is governed by a 2D advection-diffusion partial differential equation (PDE), and the plume front is modelled by a level set with a pre-specified threshold value. Distributed consensus Luenberger-type PDE observers are first constructed using local concentration measurements from the RSNs for unanimous estimate of the dynamic concentration field over the entire spatial domain. With the aid of the distributed consensus observers, a leaderless cooperative control scheme is then developed for the RSNs to monitor the dynamic plume front expansion. It is proved using the Lyapunov stability method and set stability concept that the proposed cooperative control scheme guarantees tracking of the dynamic plume front expansion and coverage of the plume front. Extensive numerical simulation results demonstrate the effectiveness and merit of the proposed cooperative control scheme.

Simulation of Diffusion Advection Phenomena with Retardation and Source/Sink Terms Using a Series of Completely Mixed Reactors

In this research, pollution transport in a 1-D domain with a source/sink and a retardation term is simulated using a series of completely mixed reactors. It is shown that as the volume of these reactors approaches zero, the governing equation of the system becomes identical to the advection diffusion partial differential equation. By application of mass balance to a finite number of completely mixed reactors, a set of ordinary differential equations which is equivalent to spatially discretized form of the equation is obtained in a completely natural and intuitive way. More importantly, the proposed innovative model can offer a conceptual insight into pollution transport phenomena and is flexible to any change for different problems.

Control of multi-agent systems with input delay via PDE-based method

This paper deals with the control of collective dynamics of a large scale multi-agent system (MAS) moving in a 3-D space under the occurrence of arbitrarily large boundary input delay. The collective dynamics is described by a pair of reaction–advection–diffusion partial differential equations (PDEs) consisting of a complex-valued state whose real-part and imaginary-part denote the position coordinates x and y, respectively, and a real-valued state equation governing the evolution of the collective dynamics in the z coordinate. A 2-D cylindrical surface represents the indexes in a continuum and defines the topology where the agents communicate with each other based on a leader–follower strategy. The agents located at the boundary are chosen as the leaders and are subject to input delays that are practically induced by communication, measurement or even actuator constraints. A boundary control, which compensates the effect of the delay and drives all the agents to the desired formation is designed based on PDE-backstepping method. Here, the 2-D cylindrical coordinate leads to the existence of singular points in the gain kernels, which makes the stability analysis non-trivial in comparison to the 1-D problem. The proposed controller ensures the exponential stability of the MAS in H2 norm under full-state measurement and transitions from one formation to another are achievable as illustrated by the simulation results.